Brief PapersContainment analysis and design for general linear multi-agent systems with time-varying delays☆
Introduction
Containment control of multi-agent systems has received considerable attention from scientific community in the past three years. In containment control problems, agents are classified into leaders and followers. The aim of containment control is to ensure that the states/outputs of followers converge to the convex hull formed by the states/outputs of leaders. Containment control has broad potential applications in various areas [1]. For example, in the scenario that a group of wheeled robots migrate in a hazardous environment, a small part of the robots equipped with necessary sensors to detect the hazards can be designated as leaders while the rest without sensors can be designated as followers. By ensuring that the followers stay in the safety area detected by the leaders using the containment control approach, the whole robots can move to the destination safely with a much less cost [2]. Containment control essentially is a kind of tracking problem. Different from the consensus tracking problems discussed in [3], [4], [5], [6], [7], [8], there can be more than one leaders in containment problem. Consensus tracking problem can be regarded a special case of containment problem where only one leader exists. Therefore, containment control has more generality.
Pioneering work on containment control problems for multi-agent systems can be found in [9], [10]. Ji et al. [11] proposed a hybrid “stop-go” policy for first-order multi-agent systems to achieve containment. Notarstefano et al. [12] addressed containment problems for first-order multi-agent systems with undirected switching topologies. Necessary and sufficient conditions for first-order and second-order multi-agent systems to achieve containment can be found in [13], [14], respectively. Lou and Hong [15] discussed containment problems for second-order multi-agent systems with randomly switching topologies. Wang et al. [16] considered the effects of communication noise on the containment control of first-order and second-order multi-agent systems. Containment control problems for second-order multi-agent systems with time-varying delays were investigated in [17]. Using only the position information, Zhang et al. [18] constructed a protocol with saturations and proposed sufficient conditions for second-order multi-agent systems to achieve containment in finite time. Robust containment control problems for second-order multi-agent systems with nonlinear dynamics were investigated in [19]. Distributed reference model based containment control problems for second-order multi-agent systems were addressed in [20].
The dynamics of each agent in [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] is assumed to be first-order or second-order. In practical applications, the dynamics of each agent is of high order. Liu et al. [21] studied state containment problems for general high-order linear time-invariant (LTI) multi-agent systems. Li et al. [22] proposed sufficient conditions for continuous-time and discrete-time general high-order multi-agent systems to achieve state containment. Dong et al. [23] investigated state containment problems for general high-order LTI singular multi-agent systems with constant time delays. Wen et al. [24] studied robust containment problems for general multi-agent systems with uncertainties using a non-smooth control approach. The effects of bounded exogenous disturbance were considered in [25]. Li et al. [26] studied state containment problems for multi-agent systems with multiple leaders of bounded inputs using distributed continuous controllers. Both static and dynamic output feedback based protocols were constructed for multi-agent systems to achieve state containment in [27]. Necessary and sufficient conditions for general high-order LTI multi-agent systems to achieve output containment were presented in [28]. It is well-known that in practical applications, especially in practical networked systems, time-varying delays are inevitable due to that the communication bandwidth and the transmission rate are limited (see e.g., [29], [30], [31], [32] and the references therein). However, time-varying delays were not considered in [21], [22], [23], [24], [25], [26], [27], [28].
In this paper, containment problems for general high-order LTI multi-agent systems with time-varying delays are investigated. Firstly, a state-feedback protocol with time-varying delays is constructed, where the motion modes of leaders can be specified. Then using integral inequality and Lyapunov–Krasovskii functional approaches, sufficient conditions for multi-agent systems to achieve containment are proposed in terms of linear matrix inequalities (LMIs). Finally, an approach to determine the gain matrices in the protocol is presented by solving four LMIs independent of the number of agents. Compared with the previous work on containment, the contributions of this paper are threefold. Firstly, the dynamics of each agent is general high-order. In [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], the dynamics of each agent is first-order or second-order. Although containment problems for second-order multi-agent systems with time-varying delays were discussed in [17], the results in [17] cannot be extended to solve the containment problems in the current paper as the general high-order dynamics does not have the special structure addressed in [17]. Since first-order and second-order dynamics are special cases of general high-order ones, the problems in the current paper are more general. Secondly, the effects of time-varying delays on containment control of general high-order LTI multi-agent systems are considered in this paper. In [21], [22], [23], [24], [25], [26], [27], [28], the dynamics of each agent is high-order but none of them considered time-varying delays. Thirdly, both sufficient conditions for general high-order multi-agent systems with time-varying delays to achieve containment and approaches to design the protocol are proposed. In the case where only one leader exists, the obtained results can also be applied to solve the consensus tracking problems for general high-order LTI multi-agent systems with time-varying delays.
The rest of this paper is organized as follows. In Section 2, some useful concepts on graph theory are introduced, and the containment control problem is formulated. In Section 3, sufficient conditions for multi-agent systems with time-varying delays to achieve containment are proposed, and an approach to design the protocol is presented. A numerical example is given in Section 4. Finally, conclusions are drawn in Section 5.
Throughout this paper, for simplicity of notation, let 0 be a zero matrix of appropriate size with zero vector and zero number as special cases. Denote by I and an identity matrix with appropriate dimension and the Kronecker product, respectively. Asterisk (⁎) represents a term which is induced by symmetry in symmetric block matrices. The superscripts H and T represent the Hermitian adjoint and the transpose of a matrix, respectively.
Section snippets
Preliminaries and problem description
In this section, some basic concepts on graph theory are reviewed and the problem description is given.
The interaction topology of a multi-agent system can be represented by a directed graph G denoted by , where is the node set, is the edge set and is the weighted adjacency matrix with . Denote by the edge in G. Define if and only if and otherwise. The neighbor set of node is represented by
Main results
In this section, firstly, sufficient conditions for multi-agent system (1) with protocol (3) to achieve state containment are proposed in terms of LMIs. Then an approach to determine the gain matrices in protocol (3) is given. Assumption 2 For each follower, there exists at least one leader that has a directed path to it. Remark 3 Assumption 2 implies that all follows should be affected directly or indirectly by at least one leader. Since containment problems can also be regarded as tracking problems with multiple
Numerical simulations
In this section, a numerical example is given to illustrate the effectiveness of the theoretical results obtained in the previous sections.
Consider a third-order multi-agent system with five followers and four leaders. The directed interaction topology among leaders and followers is shown in Fig. 1. For simplicity, it is assumed that G is 0–1 weighted. The dynamics of each agent is described by (1) with and
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Conclusions
Containment control problems for general high-order LTI multi-agent systems with time-varying delays were investigated. A protocol was constructed by the state information of each agent and neighboring agents, where the motion modes of leaders can be specified. Using Lyapunov–Krasovskii functional approach, sufficient conditions for multi-agent systems with time-varying delays to achieve containment were proposed which only include four linear matrix inequalities independent of the number of
Xiwang Dong received his BE degree in Automation from Chongqing University, Chongqing, China, in 2009, and PhD degree in Control Science and Engineering from Tsinghua University, Beijing, China, in 2014. He is now a Lecturer with the School of Automation Science and Electronic Engineering, Beihang University, Beijing, China. His research interests include consensus control, formation control and containment control of multi-agent systems. He is the recipient of the Academic Rookie Award in
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Xiwang Dong received his BE degree in Automation from Chongqing University, Chongqing, China, in 2009, and PhD degree in Control Science and Engineering from Tsinghua University, Beijing, China, in 2014. He is now a Lecturer with the School of Automation Science and Electronic Engineering, Beihang University, Beijing, China. His research interests include consensus control, formation control and containment control of multi-agent systems. He is the recipient of the Academic Rookie Award in Department of Automation, Tsinghua University in 2014, Outstanding Doctoral Dissertation Award of Tsinghua University in 2014 and Springer Thesis Award in 2015.
Liang Han was born in 1989 in Harbin, China. He received the BE degree in Automation from Nanjing University of Science and Technology, Nanjing, China, in 2011. From 2013 to 2014, he was a Research Scholar in the Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA. Currently, he is a Doctoral Student in the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include adaptive control, formation control and formation-containment control.
Qingdong Li received his BE degree in Automation from Northwestern Polytechnical University, Xi׳an, China, in 2001, ME and PhD degrees in Marine Engineering from Northwestern Polytechnical University, Xi׳an, China, in 2004 and 2008, respectively. Since 2009, he has been a Lecturer with the School of Automation Science and Electronic Engineering, Beihang University, Beijing, China. His research interests include aircraft guidance, navigation and control, fault detection, isolation and recovery, and cooperative control of multi-agent systems.
Jian Chen received his BE degree in Electronic Engineering and Automation and PhD degree in Guidance, Navigation and Control from Beihang University, Beijing, China, in 2004 and 2011, respectively. He is currently a Postdoctoral Research Fellow with the School of Automation Science and Electronic Engineering, Beihang University, Beijing, China. His current research interests include integrated guidance and control of aircrafts, and cooperative control of multi-agent systems.
Zhang Ren received his BE, ME and PhD degrees in Aircraft Guidance, Navigation, and Simulation from Northwestern Polytechnical University, Xi׳an, China, in 1982, 1985 and 1994, respectively. From 1995 to 1998, he was a Professor in School of Marine Engineering, Northwestern Polytechnical University, Xi׳an, China. From 1999 to 2000, he held the Visiting Professor positions with University of California, Riverside, and Louisiana State University, USA, respectively. He is now a Professor with the School of Automation Science and Electronic Engineering, Beihang University, Beijing, China and also the recipient of the Chang Jiang Professorship awarded by the Education Ministry of China. His research interests include aircraft guidance, navigation and control, fault detection, isolation and recovery, and cooperative control of multi-agent systems.
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This work was supported by the National Natural Science Foundation of China and Fundamental Research Funds for the Central Universities under Grants 61503009, 61374034, 61333011 and YWF-14-RSC-101.